Theorem mulsrpr 9456

Description: Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
mulsrpr	|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )

Proof of Theorem mulsrpr

Dummy variables x y z w v u t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxpi 5021	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> <. A , B >. e. ( P. X. P. ) )
2		enrex 9447	. . . . 5 |- ~R e. _V
3	2	ecelqsi 7369	. . . 4 |- ( <. A , B >. e. ( P. X. P. ) -> [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
4	1, 3	syl 16	. . 3 |- ( ( A e. P. /\ B e. P. ) -> [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
5		opelxpi 5021	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> <. C , D >. e. ( P. X. P. ) )
6	2	ecelqsi 7369	. . . 4 |- ( <. C , D >. e. ( P. X. P. ) -> [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
7	5, 6	syl 16	. . 3 |- ( ( C e. P. /\ D e. P. ) -> [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
8	4, 7	anim12i 566	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) ) )
9		eqid 2443	. . . 4 |- [ <. A , B >. ] ~R = [ <. A , B >. ] ~R
10		eqid 2443	. . . 4 |- [ <. C , D >. ] ~R = [ <. C , D >. ] ~R
11	9, 10	pm3.2i 455	. . 3 |- ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R )
12		eqid 2443	. . 3 |- [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R
13		opeq12 4204	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14	13	eceq1d 7350	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> [ <. w , v >. ] ~R = [ <. A , B >. ] ~R )
15	14	eqeq2d 2457	. . . . . . 7 |- ( ( w = A /\ v = B ) -> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R <-> [ <. A , B >. ] ~R = [ <. A , B >. ] ~R ) )
16	15	anbi1d 704	. . . . . 6 |- ( ( w = A /\ v = B ) -> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) <-> ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) ) )
17		simpl 457	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> w = A )
18	17	oveq1d 6296	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( w .P. C ) = ( A .P. C ) )
19		simpr 461	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> v = B )
20	19	oveq1d 6296	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( v .P. D ) = ( B .P. D ) )
21	18, 20	oveq12d 6299	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( ( w .P. C ) +P. ( v .P. D ) ) = ( ( A .P. C ) +P. ( B .P. D ) ) )
22	17	oveq1d 6296	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( w .P. D ) = ( A .P. D ) )
23	19	oveq1d 6296	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( v .P. C ) = ( B .P. C ) )
24	22, 23	oveq12d 6299	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( ( w .P. D ) +P. ( v .P. C ) ) = ( ( A .P. D ) +P. ( B .P. C ) ) )
25	21, 24	opeq12d 4210	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. = <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. )
26	25	eceq1d 7350	. . . . . . 7 |- ( ( w = A /\ v = B ) -> [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )
27	26	eqeq2d 2457	. . . . . 6 |- ( ( w = A /\ v = B ) -> ( [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R <-> [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) )
28	16, 27	anbi12d 710	. . . . 5 |- ( ( w = A /\ v = B ) -> ( ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) <-> ( ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) ) )
29	28	spc2egv 3182	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> E. w E. v ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) ) )
30		opeq12 4204	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
31	30	eceq1d 7350	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> [ <. u , t >. ] ~R = [ <. C , D >. ] ~R )
32	31	eqeq2d 2457	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> ( [ <. C , D >. ] ~R = [ <. u , t >. ] ~R <-> [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) )
33	32	anbi2d 703	. . . . . . 7 |- ( ( u = C /\ t = D ) -> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) <-> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) ) )
34		simpl 457	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> u = C )
35	34	oveq2d 6297	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( w .P. u ) = ( w .P. C ) )
36		simpr 461	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> t = D )
37	36	oveq2d 6297	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( v .P. t ) = ( v .P. D ) )
38	35, 37	oveq12d 6299	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( ( w .P. u ) +P. ( v .P. t ) ) = ( ( w .P. C ) +P. ( v .P. D ) ) )
39	36	oveq2d 6297	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( w .P. t ) = ( w .P. D ) )
40	34	oveq2d 6297	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( v .P. u ) = ( v .P. C ) )
41	39, 40	oveq12d 6299	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( ( w .P. t ) +P. ( v .P. u ) ) = ( ( w .P. D ) +P. ( v .P. C ) ) )
42	38, 41	opeq12d 4210	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. = <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. )
43	42	eceq1d 7350	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R )
44	43	eqeq2d 2457	. . . . . . 7 |- ( ( u = C /\ t = D ) -> ( [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R <-> [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) )
45	33, 44	anbi12d 710	. . . . . 6 |- ( ( u = C /\ t = D ) -> ( ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) <-> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) ) )
46	45	spc2egv 3182	. . . . 5 |- ( ( C e. P. /\ D e. P. ) -> ( ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) -> E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
47	46	2eximdv 1699	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> ( E. w E. v ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. C ) +P. ( v .P. D ) ) , ( ( w .P. D ) +P. ( v .P. C ) ) >. ] ~R ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
48	29, 47	sylan9 657	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( ( ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
49	11, 12, 48	mp2ani 678	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )
50		ecexg 7317	. . . 4 |- ( ~R e. _V -> [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R e. _V )
51	2, 50	ax-mp 5	. . 3 |- [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R e. _V
52		simp1 997	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> x = [ <. A , B >. ] ~R )
53	52	eqeq1d 2445	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( x = [ <. w , v >. ] ~R <-> [ <. A , B >. ] ~R = [ <. w , v >. ] ~R ) )
54		simp2 998	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> y = [ <. C , D >. ] ~R )
55	54	eqeq1d 2445	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( y = [ <. u , t >. ] ~R <-> [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) )
56	53, 55	anbi12d 710	. . . . . 6 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) <-> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) ) )
57		simp3 999	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )
58	57	eqeq1d 2445	. . . . . 6 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R <-> [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )
59	56, 58	anbi12d 710	. . . . 5 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) <-> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
60	59	4exbidv 1705	. . . 4 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) -> ( E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) <-> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
61		mulsrmo 9454	. . . 4 |- ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )
62		df-mr 9439	. . . . 5 |- .R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) }
63		df-nr 9437	. . . . . . . . 9 |- R. = ( ( P. X. P. ) /. ~R )
64	63	eleq2i 2521	. . . . . . . 8 |- ( x e. R. <-> x e. ( ( P. X. P. ) /. ~R ) )
65	63	eleq2i 2521	. . . . . . . 8 |- ( y e. R. <-> y e. ( ( P. X. P. ) /. ~R ) )
66	64, 65	anbi12i 697	. . . . . . 7 |- ( ( x e. R. /\ y e. R. ) <-> ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) )
67	66	anbi1i 695	. . . . . 6 |- ( ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) <-> ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) )
68	67	oprabbii 6337	. . . . 5 |- { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) } = { <. <. x , y >. , z >. | ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) }
69	62, 68	eqtri 2472	. . . 4 |- .R = { <. <. x , y >. , z >. | ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) }
70	60, 61, 69	ovig 6409	. . 3 |- ( ( [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R e. _V ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) )
71	51, 70	mp3an3 1314	. 2 |- ( ( [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) )
72	8, 49, 71	sylc 60	1 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 974   = wceq 1383   E.wex 1599   e. wcel 1804   _Vcvv 3095   <.cop 4020   X. cxp 4987  (class class class)co 6281   {coprab 6282   [cec 7311   /.cqs 7312   P.cnp 9240   +P. cpp 9242   .P. cmp 9243   ~R cer 9245   R.cnr 9246   .R cmr 9251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-omul 7137  df-er 7313  df-ec 7315  df-qs 7319  df-ni 9253  df-pli 9254  df-mi 9255  df-lti 9256  df-plpq 9289  df-mpq 9290  df-ltpq 9291  df-enq 9292  df-nq 9293  df-erq 9294  df-plq 9295  df-mq 9296  df-1nq 9297  df-rq 9298  df-ltnq 9299  df-np 9362  df-plp 9364  df-mp 9365  df-ltp 9366  df-enr 9436  df-nr 9437  df-mr 9439

This theorem is referenced by:  mulclsr  9464  mulcomsr  9469  mulasssr  9470  distrsr  9471  m1m1sr  9473  1idsr  9478  00sr  9479  recexsrlem  9483  mulgt0sr  9485